proton–proton and proton–antiproton differential elastic …...elastic scattering and...

Turk J Phys(2017) 41: 426 – 435© TÜBİTAKdoi:10.3906/fiz-1704-9

Turkish Journal of Physics

http :// journa l s . tub i tak .gov . t r/phys i c s/

Research Article

Proton–proton and proton–antiproton differential elastic cross sections modelingat high and ultra-high energies using a hybrid computing paradigm

EL-Sayed EL-DAHSHAN1,2,∗1Department of Physics, Faculty of Sciences, Ain Shams University, Abbassia, Cairo, Egypt

2Egyptian E-Learning University (EELU), Eldoki, El-Geiza, Egypt

Received: 10.04.2017 • Accepted/Published Online: 18.08.2017 • Final Version: 10.11.2017

Abstract: This work presents a hybrid computing technique for modeling the differential elastic cross section of bothproton–proton “pp” and proton–antiproton “pp(bar)” collisions from high to ultra-high energy regions (from 13.9 GeV to14 TeV) as a function of the center-of-mass energy “s” squared and four momentum transfer squared “t”. We proposed agenetic algorithm (GA) and support vector regression (SVR) hybrid techniques to calculate and predict the “differentialelastic cross section” of both “pp” and “pp(bar)”. Our proposed GA-SVR hybrid model shows a good match to theavailable experimental data, as well as predicting the latest and future “TOTEM” experiments for differential elasticcross sections that are not utilized in the training phase. An extrapolation of the differential cross section to |t| →0 givesthe total cross section through the optical theorem. The model calculations and predictions for “pp” and “pp(bar)” totalcross-sections over a wide collision energy ranging from ISR to LHC are given. All the results for the total cross-sectionsmatched well with the last and the more recent measurements (

√s= 7 TeV and 8 TeV), as well as the results of other

models for the future “TOTEM” measurements at (√s = 10 TeV and 14 TeV).

Key words: Deferential elastic cross sections, proton–proton (antiproton) collision, empirical modeling, support vectorregression, genetic algorithm

1. IntroductionProton–proton (pp) and proton-antiproton ( PP ) cross sections at high energy are the first studies performedwhen a new hadron accelerator opens up a new energy region. The study of the total PP and PP cross sectionand its various subcomponents (elastic, inelastic, and diffractive) is a very powerful tool to understand thestrong interaction mechanism [1–6]. Elastic pp and pp̄ scattering is the most essential component in highenergy hadronic interactions to investigate the structure of the proton [7]. There are three important physicalcomponents to characterize the elastics pp and pp̄ scattering: total cross section “σpp, pptot ”, differential cross

section “(dσdt

)pp, pp̄ ”, and the ratio of the forward real to imaginary parts of the elastic scattering amplitude“ρ -parameter” [8]. The elastic differential cross section of PP and PP contains information about protonstructure and nonperturbative aspects of PP and PP interactions, as well as providing useful information onthe behavior of the scattering amplitude, in particular, on the nature of their diffractive components [9]. The“total cross section” σpp, pptot can be obtained from the calculation of elastic scattering ones depending on the∗Correspondence: [email protected]

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“optical theorem” [10,11]. The investigation of high energy PP and PP elastic cross section depends mainly onthe “center of mass energy” (

√s) and the “four-momentum transfer squared” (t) [1–5].

The high energy PP and PP elastic scattering have been measured experimentally at different collidersover a wide range of

√s, such as “Intersecting Storage Rings (ISR)” and “Super Proton Synchrotron (SPS)”

colliders at the “European Council for Nuclear Research (CERN)” [12–20], as well as the “Fermilab” fixed targetexperiments and Tevatron colliders [21–29]. Recently, the measurements of the high energy PP and PP elasticscattering have been working at the Large Hadron Collider “LHC” by the “TOTEM” (TOTal cross section,Elastic scattering and diffraction dissociation Measurement) at the LHC Collaboration [30–35]. Furthermore,different theoretical models have been proposed to investigate the PP and PP elastic scattering; most ofthem are QCD-based models [9]. This wide array of experimental and theoretical models attempt to describepp elastic scattering at LHC. In addition, the understanding of this process will provide fundamental insightinto the nonperturbative and the perturbative QCD [7,9]. To our knowledge, despite the amount of availableexperimental data and model descriptions of these data, a precise and complete description of differential andtotal cross sections is still lacking and this issue is an open issue. Considering that, we hope our model maybring insights for further and deeper developments.

In recent years, the artificial neural networks (ANNs) [36] approach has been widely used for predictionof physics as classification and regression problems [37–42]. However, this approach has been found to be moreefficient compared to statistical algorithms. It has many downsides such as over-fitting, slow convergence, andless generalizing performance. Support vector machine (SVM) is a machine-learning algorithm developed byVapnik (1995) to solve regression problems [43]. The SVMs also have been modified to the so-called supportvector regression (SVR) for manipulating nonlinear regression estimation problems [43–45]. Thus, the processof applying SVMs in regression problems is referred to as SVR. However, it is necessary to optimize the SVMparameters. Appropriate parameters in SVR lead to overfitting or underfitting. Therefore, selecting optimalhyperparameters is an important step in SVR design. Genetic algorithms (GAs) are search algorithms basedupon the mechanics of natural selection [46,47]. In this study, we propose a GA to determine parameters ofSVR that optimizes all SVR’s parameters simultaneously from the training data. Thus, the aim of this article

is to suggest a hybrid model GA-SVR to calculate and predict the “(dσdt

)pp, pp ” as a function of “√s” and “ |t|”,as well as “σpp, pptot ” at various

√s ranging from nearly

√s = 13.9 GeV to

√s = 14 TeV for |t| = 0 to 6 GeV2 .

The next sections of this paper are structured as follows: Section 2 presents the proposed GA-SVR tomodel the PP and PP deferential elastic cross sections. The results obtained are explained in Section 3. Lastly,Section 4 provides the findings and conclusions.

2. The proposed hybrid GA-SVR model

Creating an accurate mathematical model that can simulate and predict the nonlinear behavior of physicalphenomena is a challenging issue. In the present study, we developed a hybrid soft computing approach GA-SVR based on experimental data to calculate and predict the differential elastic cross section of both PP and PP

collisions,(dσdt

)pp, pp̄ , i.e. dσdt = f (√s, t) as well as σpp, pptot . The following subsections describe the developmentof our hybrid computing GA-SVR model for calculating and predicting of the

(dσdt

)pp, pp̄ and σpp, pptot .427


2.1. Mathematical formulation of support vector regression

SVM was invented in 1992 by Vapink et al. [43–45] as a classification tool and was extended to a nonlinearregression case “support vector regression (SVR)” in 1997 [43–45]. SVR is a powerful function approximationapproach based on statistical learning theory. Using the so-called ε -insensitive loss function for SVR (ε -SVR),we ensure the finding of the global minimum and the optimization of the generalization error over the parameterspace.

In SVR, the given training set is {(xi, yi), i = 1, ...,ℓ} ⊂ Rd × R, where xi ∈ Rd is the ith input vectorin the “input space x”, yi is the corresponding ith output (in our case x1 denotes the center of mass energy(√s) and the four – momentum transfer squared (t), yi denotes the differential cross section for PP and PP

collisions “(dσdt

)pp, pp̄ ”; the input space x is first mapped onto high-dimensional “feature space” using nonlinearmapping function (kernel function g(x)) and then a linear model is constructed in this feature space to estimatethe output yi using the SVR regression function f(x) = w g(x) + b;w and b are the “weight” vector and the“bias” term, respectively. Using mathematical notation, the generic linear SVR approximator f(x, w) in thehigh-dimensional feature space is defined as follows:

dσ

dt= f(x, w) = f

(√s, t, w

)=

∑mj=1

wigj (x) + b, (1)

where gj (x) , j = 1, ....., m denotes a set of nonlinear transformations (mapping function), and w and b arethe “weight” vector and the “bias” term, respectively.

The quality of estimation of the ε -SVR regression model is measured by a new type of loss function calledε -insensitive loss function proposed by Vapnik [43–45]. ε -SVR performs linear regression in the high-dimensionfeature space using ε -insensitive loss and, at the same time, tries to reduce model complexity by minimizing12 ∥w∥

2 (norm of the weights vector), which is subject to yi(wTxi + b) ≥ 1 . This is an optimization problem,which can be converted to the dual optimization problem. Therefore, the regression function (the estimatedfunction of the output, yi , in our case is dσdt ) “eqn 1” can be rewritten as a generic function and is given by

f(xi, α∗, α) =

nSV∑i=1

(αi − α∗i )K(xi, x) + b

0 ≤ αi, α∗i ≤ C (C is penalty parameter) andN∑i=1

(αi − α∗i ) = 0 (2)

where “nSV ” is the number of Support Vetors “SV s ”The Lagrange multipliers of the dual optimization problem, αi, α∗i are obtained and then w and b

are given as ω =l∑

i=1

(αi − α∗i )x and b = − 12 ⟨ω, (xp + xs)⟩ , where xr and xs are any support vectors and

k (xi, xj) = (g (xi) . g (xj)) is the kernel function. The kernel function used in the study is a (Gaussian kernel)radial basis function (RBF), k(xi, xj) = exp

(−γ||xi − x∗j ||2

), where xi and xj are inner vectors in the input

space and γ is a parameter that sets the “spread” of the kernel.According to the above short overview of the SVR, (C, γ, ε) are the most important parameters of the

SVR approach. All these parameters are considered for calculating and predicting the new output based on

428


any new input features and they are user-defined parameters. Various methods are proposed for selecting theseparameters automatically. In this study, the evolutionary GA is utilized to select the optimal parameters of theSVR model to achieve a better generalization performance of the hybrid (GA-εSVR) model.

2.2. Genetic algorithm approach

GAs are stochastic search techniques based on the Darwinian principle of natural selection, “survival of thefittest”, and genetics gained attention after Holland in 1975 [46]. The GA method has three basic evolutionaryoperators, “reproducing operators”: selection, crossover, and mutation. Based on these operators, the GAbegins from a randomly made up population of individuals (chromosomes/solutions), which are expressed asa “genotype”. For each generation, the fitness of chromosomes is evaluated based on a fitness function [46,47].The size of the population and the probability for crossover and mutation are called the control parametersof the GA. The solution in the real world is known as a “phenotype”. The genotype is transformed into thephenotype for the fitness evaluation process. For more information about the GA, please see Ref. [46].

2.3. Workflow of the GA-SVR modelTo build and train our GA-SVR approach for calculation and prediction of differential elastic cross section(dσdt

)pp, pp̄ , the available experimental datasets [11–35] are randomly divided into two subsets, one for trainingand the other for testing the GA-SVR model. We apply 5-fold cross-validation accuracy computing by SVR onthe training data set (experimental data) to the GA-fitness function. The training dataset is used to obtain theSVR parameters and radial basis function (RBF) width parameter (C, γ, ε) . The testing data are utilized toinvestigate the prediction performance of the model. At first, the training inputs and outputs, range of the SVRand RBF parameters, population size, and number of generations are fed to the GA. Then the initial populationis generated in the GA. Moreover, 5-fold cross validation is used for avoiding the problem of overfitting duringthe training and obtaining the best fitness values. The fitness value (mean squared error (MSE)) is obtainedfor each individual of the population. This procedure is repeated until all individuals (candidate solutions)are evaluated. After that, a new population is generated using genetic operators, i.e. selection, crossover, andmutation operators. Generation of the new populations continues for 50 times. The genetic operators, onepoint crossover, and mutation are used. The crossover rate is 90% and mutation rate is 10%. The Tournoiselection method is used to select the mating pool. The optimal SVR parameters (C, γ, ε) are obtained tolearn the model. Then the optimal parameters are chosen using the individual with the best fitness value at thelast generation. The final evaluation of the model performance is the testing based on the unseen experimentaldata. The criteria of root mean square (MSE) and coefficient of determination (R2) were used for evaluating

the performance of the GA-SVR models [48]. The developed model GA-εSVR is used to predict(dσdt

)pp, pp̄ inthis study. The basic flowchart of the GA-εSVR model is depicted in Figure 1. The proposed hybrid GA-SVRmodel was developed and implemented in the MATLAB v6.5 environment with high degree of accuracy andreliability.

3. Results and discussionWe have applied the developed hybrid computing approach using GA and ε -SVR to reasonably reproduce andcalculate the differential elastic cross section based on the existing experimental data for PP and PP scattering,as described in the above section. The available experimental data are collected from a set of accelerators in a

429


Figure 1. Flow diagram of our GA-εSVR model.

wide energy range from ISR energy range up to the LHC, including 14, 23.5. 30.7, 44.6, 52.8-, 62 GeV, as wellas the recent LHC energies 7 and 8 TeV (CMS preliminary) for pp scattering [11–35] and 14, 53, 546 GeV, and1.800 TeV for PP scattering [12–35]. These datasets cover the momentum transfer range 0 < t < 6 GeV2 .

The available experimental studies covering√s range from 23 to 62 GeV at ISR to 2 TeV at Tevatron, as

well as 7 and 8 TeV. At LHC, pp elastic differential cross section is planned to be measured at the unprecedented√s of 14 TeV by the experimental group TOTEM in the “ |t|” range 0 > t > 6 GeV2 .

Figure 2 (a and b) illustrates the dependence of differential elastic cross section(dσdt

)pp, pp̄ on |t| , whichshows a sharp peak at small values of |t| ranging from 0.01 to 0.5 GeV2 and after this range the differentialelastic cross section decreases exponentially to a dip at about |t| = 1.4 GeV2 . In addition, from this figure,we note that after |t| = 1.4 GeV2 the differential elastic cross section manifests a broad maximum around|t| = 2 GeV2 and then gradually decreases. Moreover, the obtained results confirm that the strong peak atlow |t| seems to become smaller with the increase in energy; in addition, the dip moves towards lower valuesof |t| . This confirms the behavior observed from the previous different experimental and theoretical studies[11–35,49,50]. In addition, the comparison between the experimental and other theoretical models is shown inFigure 2. In this comparison, our model calculations are highly matched with the theoretical and experimentalones.

We have used our model to predict the differential elastic pp cross section at√s = 10, 13 (CMS

preliminary), and 14 TeV. These predicted values are compared with other models [49,50] as shown in Figure3. Figure 3 shows a good agreement between our model results and other theoretical ones [49,50].

For the sake of completeness, we applied the proposed GA-SVR model to the pp scattering data.We proceeded in the same manner as previously mentioned for pp scattering, using the SPS and Tevatronexperimental data. Figure 4 demonstrates the calculated differential elastic cross section of PP scattering at√s=14, 53, 546 GeV, and 1.800 TeV. For PP scattering, we obtained a similar behavior as pp scattering,

but without the pronounced dip, which was replaced instead by a broad “shoulder”. Furthermore, this figureshows a comparison between the experimental and other theoretical models. In this comparison, our modelcalculations show good agreement with both the experimental and theoretical ones. Moreover, in order to

430


Figure 2. (a and b).(dσdt

)pp of the PP as a function of the |t| at different √s .(*) our GA-SVR model, experimentaldata (o) [11–35], and (-) theoretical ones [49,50].

evaluate the performance of the proposed GA-SVR model, the R2 and MSE are calculated. Based on ourproposed GA-εSVR model, we calculated the total cross section of PP and PP ,σtot . For the calculation ofσtot , we first extrapolated the differential elastic cross section of each of the available data to the optical point( |t| → 0) (i.e. dσdt

∣∣t=0

) . Then the optical theorem is used for the calculation of PP and PP total cross section

σpp, pptot as follows:

σ2tot =16π (ℏc)2

1 + ρ2dσ

dt

∣∣∣∣t=0

, (3)

where ℏ and c are the reduced Plank’s constant and the speed of light (equal to one: c = 1 and ℏ = 1),respectively. For the ρ parameter, the COMPETE preferred-model [49] extrapolation of 0.141 ± 0.007 waschosen. Using the conversion factor, Eq. (3) becomes

σtot = 4.382

(dσ

dt

∣∣∣∣t=0

)1/2(4)

To calculate σtot from Eq. (4), we obtained dσdt∣∣t=0

for PP and PP by extrapolation of differential elasticcross sections to |t| → 0 based on our GA-SVR model for the corresponding range of energy

√s =14, 23.5.

30.7, 44.6, 52.8-, 62 GeV, 7 TeV and 8 TeV,10 TeV, 13 TeV, 14 TeV for pp and√s ; =13.9, 53, 546, 1800 GeV for

PP . A comparison is made between our results of the total cross section σtot , and the corresponding theoretical

431


Figure 3.(dσdt

)pp of the PP as a function of the |t| at√s = 10, 13, 14 TeV.

Figure 4.(dσdt

) pp̄ of the PP as a function of the |t| atdifferent

√s .(*) our GA-SVR model, experimental data

(o) [11–35] and (-) theoretical ones [49,50].

and experimental ones as shown in Figure 5. This figure shows that the proposed model gives good agreementwith the experimental [11-35] and theoretical [49-50] results.

4. Conclusions

In the present work, we developed and applied a hybrid computing approach GA-εSVR for the calculation and

prediction of the differential elastic cross sections for PP and PP scattering,(dσdt

)pp, pp̄ , as well as the totalcross sections σpp, pptot at various center of mass energies

√s ranging from nearly 13.9 GeV to 14 TeV in the

transfer momentum squared range t = 0 to 6 GeV2 . A large data set from different labs was used to train andtest the model. The proposed model reproduced well the trained experimental differential elastic cross sections,as well as the unutilized data during the training phase. Good agreement between the model predictions throughthe higher energies to be reached at future LHC experiment at 10 TeV, 13 TeV, and 14 TeV was obtained, aswell as with other theoretical models. After the extrapolation of the available data of the differential elasticcross section in the low “t” range

(i. e. dσdt

∣∣t→0

), a total cross section for PP and PP , σpp, pptot , was obtained by

applying the optical theorem. The common behavior of the obtained results and the predicted ones of σpp, pptotshowed a good match with the latest LHC measurements “TOTEM experiments” and the predicted ones byother models. The results of this model’s calculations are very encouraging and give rise to the hope that furtherstudies into pp and PP scattering can be conducted to describe all the particle production observables.

432


Figure 5. The total PP and PP cross section “σpp, pptot ”.

5. Acknowledgments

The author would like to thank the Editor-in-Chief and the anonymous referees for their helpful comments andfor the kind suggestions that improved the manuscript seriously.

References

[1] Siacomelli, G. Phys. Rep. 1976, 23, 123-235.�

[2] Menon, M. J. Brazn J. Phys. 2005, 35,100-121.�

[3] Austin, D. M.; Greiman, W. H.; Rarita, W. Phys. Rev. D 1970, 2, 2613-2627.

[4] Block M. M.; Cahn R. N. Rev. Mod. Phys. 1985, 57, 563-598.

[5] Cartiglia, N. arXiv: hep-ex /1305.6131 (2013).

[6] Csorgo, T.; Glauber, R. J. arXiv:hep-ph / 1311.2308 (2013).

[7] Martini, A. F. Phys. Rev. D, 1997, 56, 4338-4349.

[8] Martynov, E. Phys. Rev. D , 2013, 87, 114018-1-114018-8.

[9] Dokshitzer, Y. Basics of Perturbative QCD. Atlantica Séguier Frontières, 1991.

[10] Kohara, A. K.; Ferreira, E.; Kodama, T. Eur. Phys. Jour. C 2014, 74, 3175.

[11] Fagundes, D. A.; Grau, A.; Pacetti, S., Pancheri, G.; Srivastava, Y. N. arXiv:hep-ph / 1307.0298 (2013).

[12] Akerlof, C. W.; Kotthaus, R.; Loveless, R. L.; Meyer, D. I.; Ambats, I.; Meyer, W. T.; Yovanovitch, D. D. Phys.Rev. D 1976, 14, 2864-2877.

433


[13] Breakstone, A., Crawley, H. B., Dallavalle, G. M., Doroba, K., Drijard, D., Fabbri, F.; Giacomelli, G. Phys. Rev.Lett. 1985, 54, 2180-2183.

[14] Arnison, G.; Astbury, A.; Aubert, B.; Bacci, C.; Bauer, G.; Bezaguet, A.; Calvetti, M. Phys. Lett. B 1983, 128,336-342.

[15] Amaldi, U.; Schubert, K. R. Nucl. Phys. B 1980,166, 301-320.

[16] Kwak, N.; Lohrmann, E.; Nagy, E.; Regler, M.; Schmidt-Parzefall, W.; Schubert, K. R.; Niebergall, F. Phys. Lett.B 1975, 58, 233-236.

[17] Amaldi, U.; Cocconi, G.; Diddens, A. N.; Dobinson, R. W.; Dorenbosch, J.; Duinker, W.; Baroncelli, A. Phys. Lett.B 1977, 66, 390-394.

[18] Baksay, L.; Baum, L., Boehm, A., Derevshikov, A., De Zorzi, G.; Giesen, H. J.; Naroska, B.; Nucl. Phys. B 1978,141, 1-28.

[19] Nagy, E.; Orr, R. S.; Schmidt-Parzefall, W.; Winter, K.; Brandt, A.; Büsser, F. W.; Schubert, K. R. Nucl. Phys.B 1979, 150, 221-267.

[20] Battiston, R.; Bozzo, M.; Braccini, P. L.; Carbonara, F.; Carrara, R.; Castaldi, R.; Koene, B. Phys. Lett. B 1983,127, 472-475.

[21] Kaidalov, A. B.; Poghosyan, M. G. arXiv:hep-ph / 1109.3697(2011).

[22] Bernard, D.; Bozzo, M.; Braccini, P. L.; Carbonara, F.; Castaldi, R.; Cervelli, F.; Kluit, P. Phys. Lett. B 1986,171, 142-144.

[23] Faissler, W.; Gettner, M.; Johnson, J. R.; Kephart, T.; Pothier, E.; Potter, D.; Shahbazian, K. Phys. Rev. D 1981,23, 33-42.

[24] Desgrolard, P.; Giffon, M.; Predazzi, E. Z. Phys. C. Part. Fields 1994, 63, 241-252.

[25] Rubinstein, R.; Baker, W. F.; Eartly, D. P.; Klinger, J. S.; Lennox, A. J.; Kalbach, R. M.; Kaplan, D. H. Phys.Rev. D 1984, 30, 1413-1432.

[26] Bozzo, M.; Braccini, P. L.; Carbonara, F.; Castaldi, R.; Cervelli, F.; Chiefari, G.; Lanzano, S. Phys. Lett. B 1985,155, 197-202.

[27] Abe F.; Albrow M.; Amid D.; Anway-Wiese C.; Apollinari G.; Atac M.; Auchincloss P.; Ami P.; Bacchetta N.;Baden A.R.; et al. CDF Collaboration, Phys. Rev. 1994, D50, 5518-5534.

[28] Augier, C.; Bernard, D.; Bourotte, J.; Bozzo, M.; Bueno, A.; Cases, R.; Matthiae, G. Phys. Lett. B 1993, 316,448-454.

[29] Arnison, G.; Astbury, A.; Aubert, B.; Bacci, C.; Bauer, G.; Bezaguet, A.; Bock, R.; Bossart, R.; Bosser, J.;Bowcock, T. J. V.; et al. UA1 Collaboration: Phys. Lett. 1983, B128, 336-342.

[30] Antchev, G.; Aspell, P., Atanassov, I.; Avati, V.; Baechler, J.; Berardi, V.; Cafagna, F. S. Phys. Let. 2011, 95,41001-p1 41001-p7.

[31] Grafstrom, P. Int. J. Mod. Phys. A 2015, 30, 1542007-1-1542007-24.

[32] Stenzel, H. arXiv: hep-ex/1611.02454 (2016).

[33] Block, M. M.; Halzen, F. Phys. Rev. D 2011, 83, 077901-1-077901-3.

[34] Islam, M. M.; Kašpar, J.; Luddy, R. J. Mod. Phys. Lett. A 2009, 24, 485-496.

[35] Nakamura, K. Particle Data Group. J. Phys. G Nucl. Partic. 2010, 37, 075021 (1422 pp).

[36] Haykin, S. Neural Networks and Learning Machines, Prentice Hall: Upper Saddle River, NJ, USA, 2008.

[37] Kiesling, C.; Zimmermann, J. (2004, December). Statistical learning methods in high-energy-and astrophysics.In Intelligent Sensors, Sensor Networks and Information Processing Conference, 2004. Proceedings of the 2004 IEEE,pp. 325.

[38] Schmidt, M.; Lipson, H. Science 2009, 324, 81-85.

434


[39] Whiteson, S.; Whiteson D. Eng. Appl. Artif. Intel. D 2009, 22, 1203-1217.

[40] Teodorescu, L. IEEE T Nucl. Sci. 2006, 53, 2221-2227.

[41] Akkoyun, S.; Bayram, T.; Turker, T. Radiat. Phys. Chem. 2014, 96, 186-189.

[42] El-Bakry, M. Y.; El-Dahshan, E. S. A.; El-Bakry, S. Y. Europ. Phys. J. Plus 2015, 130, 1-7.

[43] Cortes, C.; Vapnik, V. Mach. Learn. 1995, 20, 273-297.

[44] Vapnik V. N. Statistical Learning Theory. Wiley: New York, NY, USA, 1998.

[45] Vapnik, V. The Nature of Statistical Learning Theory. Springer: Berlin, Germany, 2013.

[46] Holland, J. H. Adaptation in Natural and Artificial Systems. An Introductory Analysis with Application to Biology,Control, and Artificial Intelligence. University of Michigan Press: Ann Arbor, MI, USA, �1975.

[47] Goldberg, D. E.; Holland, J. H. Mach. Learn. 1988, 3, 95-99.

[48] Bohm, G.; Zech, G. Introduction to Statistics and Measurement Analysis for Physicists. DESY: Hamburg Germany,2010.

[49] Bourrely, C.; Soffer, J.; Wu, T. T. Eur. J. Phys. 2003, C 28, 97-105.

[50] Cudell, J. R.; Ezhela, V. V.; Gauron, P.; Kang, K.; Kuyanov, Y. V.; Lugovsky, S. B. B.; Nicolescu, E.A.; Razuvaev,N. P. Phys. Rev. Lett. 2002, 89, 201801-1-201801-4.

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IntroductionThe proposed hybrid GA-SVR modelMathematical formulation of support vector regressionGenetic algorithm approachWorkflow of the GA-SVR model

Results and discussionConclusionsAcknowledgments

proton–proton and proton–antiproton differential elastic …...elastic scattering and...

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